Some Thermodynamic and Quantum Aspects of NMR Signal Detection Stanislav Sýkora URL of this document: www.ebyte.it/stan/Talk_BFF6_2011.html DOI: 10.3247/SL4nmr11.002 Presented at 6th BFF on Magnetic Resonance Microsystem , Saig/Titisee (Freiburg), Germany, 26-29 July 2011 Controversies about the nature of NMR signal and its most common modes of detection There is a growing body of literature showing that the foundations of Magnetic Resonance are not as well understood as they should be Principle articles: Bloch F., Nuclear Induction , Phys.Rev. 70, 460-474 (1946). Dicke R.H., Coherence in Spontaneous Radiation Processes , Phys.Rev. 93, 99-110 (1954). Hoult D.I., Bhakar B., NMR signal reception: Virtual photons and coherent spontaneous emission , Concepts Magn. Reson. 9, 277-297 (1997). Hoult D.I., Ginsberg N.S., The quantum origins of the free induction decay and spin noise , J.Magn.Reson. 148, 182-199 (2001). Jeener J., Henin F., A presentation of pulsed nuclear magnetic resonance with full quantization of the radio frequency magnetic field , J.Chem.Phys. 116, 8036-8047 (2002). Hoult D.I., The origins and present status of the radio wave controversy in NMR , Concepts in Magn.Reson. 34A, 193-216 (2009). Stan Sykora 6th BFF 2011, Freiburg, Germany Controversies about the nature of NMR signal and its most common modes of detection This author’s presentations: Magnetic Resonance in Astronomy: Feasibility Considerations , XXXVI GIDRM, 2006. Perpectives of Passive and Active Magnetic Resonance in Astronomy, 22nd NMR Valtice 2007. Spin Radiation, remote MR Spectroscopy, and MR Astronomy , 50th ENC, 2009. Signal Detection: Virtual photons and coherent spontaneous emission , 18th ISMRM, 2010. The Many Walks of Magnetic Resonance: Past, Present and Beyond, Uni Cantabria, Santander, 2010. Do we really understand Magnetic Resonance? , MMCE, 2011. Slides and more info: see www.ebyte.it/stan/SS_Lectures.html Stan Sykora 6th BFF 2011, Freiburg, Germany Questions, questions, questions … for which there should exist simple answers, but … do they? (we have no time for these points now, but I have a handout and we can discuss them during the meeting) 1) Why do we use different explanations for different aspects of MR ? 2) Which aspects of MR are undeniably quantum and cannot be described classically ? 3) Are Bloch equations classical? Is it sensible to try derive them from quantum theory ? 4) Is electromagnetic radiation involved in magnetic resonance? If so, is it true or virtual ? 5) Does spontaneous emission from spin systems occur? If so, what are its properties ? 6) Is MR a near or a far phenomenon? Is remote excitation and/or detection possible ? 7) Can an FID be described as coherent spontaneous emission ? What about CW-NMR ? 8) How do the spins interact with nearby conductors, and with the coil ? 9) Which phenomena can be described considering an isolated spin, and which can’t ? a) What is the role of relaxation processes in all this? Are they essential or marginal ? b) What is the role of time-averaged Hamiltonians in magnetic resonance ? c) Can an FID be described as a sum of individual quantum transitions ? d) Does all this uncover some gaping holes in quantum physics ? e) Does all this tell us something about the ontology of photons ? f) Can MR throw new light on basic aspects of physics ? Stan Sykora 6th BFF 2011, Freiburg, Germany We use different descriptions for different MR phenomena ! Q < 10 4 11 B 3/2 Q = ω/∆ω ≈ 10 1 M 1/2 ⊥⊥⊥ Z' ≡≡≡Z 1/2 ωn 0 -1/2 M|| M(0) -1/2 -1 M α M(t) -3/2 Y' B1 X' CLASSICAL HYBRID QUANTUM Induction law, Thermal equilibrium Coupled spins spectroscopy, Bloch equations, Theory of CW-MR Spin systems dynamics, Reciprocity theorem, Quantum coherences Hilbert/Liouville QM, Simplified MRI, Operator products Density matrix,… Stan Sykora MMCE 2011, Tatranska Lomnica, Slovakia The principle difficulty: How to reconcile the manifestly quantum aspects of MR phenomena with the totally classical nature of the detection/excitation devices Possible ways to tackle the problem A. Rely on paradigms shared by both classical and quantum physics Thermodynamics, Conservation Laws and (some) Statistical Physics A problem: the approach provides good insights but remains incomplete B. Approximate quantum spin systems in a classical way A problem: arisal of a transversal magnetization from parallel spin eigenstates C. Provide a quantum description of the detection/excitation devices A problem: apparent complexity and necessity to conform to classical results An extra task: how to couple the spin system to the detection/excitation device Stan Sykora 6th BFF 2011, Freiburg, Germany The two parts of this talk: A. Part I: Classical paradigms shared also by quantum physics Thermodynamics, Conservation Laws and (some) Statistical Physics B. Approximate quantum spin systems in a classical way A problem: everybody does this since 50 years, therefore I don’t dig it C. Part II: Quantum description of the detection/excitation devices An extra task: coupling the spin system to the detection/excitation device Stan Sykora 6th BFF 2011, Freiburg, Germany Let us forget quantum physics for a while ☺☺☺ and remember just the conservation of energy (the first law of thermodynamics) The physical system we will consider: L Loop and its Inductance R Resistance (may be distributed) J Electric current P just a Point B Magnetic Field m Magnetic Dipole B H = µ = µβ Jτττβ , where µ is the permaebility, β a constant, and τττβ a unit vector. β and τττβ depend only on the system’s geometry. m m = m τττm , where τττm is a unit vector along Stan Sykora 6th BFF 2011, Freiburg, Germany Setting up a master equation Energy terms: 2 Inductive: EL = LJ /2 2 Self-energy of m: Em = km /2 m B Interaction: Ei = - . = -κmJ , where κ = µβ (τττβ.τττm) Dissipation: dE/dt = -J2R Since R is the only dissipative element in the circuit, the energy conservation law dictates 2 dE/dt = dE L/dt + dE i/dt + dE m/dt = -J R Hence, using the apostrophe to denote time derivatives, (LJ -γm)J' - κm'J + RJ 2 +kmm' = 0 Stan Sykora 6th BFF 2011, Freiburg, Germany … a couple of useful notes … Energy terms: 2 Inductive: EL = LJ /2 2 Self-energy of m: Em = km /2 m B Interaction: Ei = - . = -γmJ , where κ = µβ (τττβ.τττm) Dissipation: dE/dt = -J2R 2 2 2 2 † Since EL = LJ /2 and also EL = ∫∫∫V|B| dτ/2 µ = µJ ∫∫∫V β dτ/2 , 2 2 we have L = µ∫∫∫V β dτ = µ<β > V showing that, upon scaling, β correlates with L1/2 The old terms coil volume and filling factor arise from here † While β is positive by definition, κ = µβ (τττβ.τττm) can be both positive or negative Stan Sykora 6th BFF 2011, Freiburg, Germany (A,B) Special cases of the master equation (LJ -γm)J' - κm'J + RJ 2 +kmm' = 0 A. When there is no magnetic dipole (m = m' = 0 ), we get 2 - (R/L) t LJJ' + RJ = 0 ⇒ LJ' + RJ = 0 ⇒ J = J 0 e This is the well-known equation of the LR-circuit . B. The term kmm' is the power needed to maintain the dipole. This is null in the case of quantum particles . It drops out also when the dipole is associated with a permanent magnet or when it is driven driven by an external device. In such cases k=0 and we have a modified master equation: LJJ' - κmJ' - κm'J + J 2R = 0 Stan Sykora 6th BFF 2011, Freiburg, Germany (C) Special cases of the modified master equation LJJ' - κmJ' - κm'J + J 2R = 0 How must the dipole evolve to maintain a constant current ? J’ = 0 The answer is that m must grow linearly with time at the rate m' = JR/ κ One can reverse the point of view and say that this is as though the linearly increasing dipole generated in the circuit the emf Vemf = JR = κm' = µβ (τττβ.τττm) m' which coincides exactly with the classical law of induction . It is easy to show that this holds also when J’≠ 0, provided the induction law is properly extended ( LJ = κm = - Φ). Stan Sykora 6th BFF 2011, Freiburg, Germany (D-E) Oscillating magnetic dipoles LJJ' - κmJ' - κm'J + J 2R = 0 D. What if the dipole is made to oscillate? Let m = m 0 exp(i ωt) and assume a solution of the type J = J 0 exp[i( ωt+ ϕ)]. Then the master equation gives J0 = 2 κω m0/|R+i ωL| and tan( ϕ) = R/ ωL. E. The amplitude of the detected current (signal) is proportional to κ and therefore also to β. Since β describes the efficiency B/J of a current J to generate the field B at point P, this proves the reciprocity theorem : The larger is the field B generated at a point P by a current J in the loop, the stronger is the detected signal due to an oscillating dipole located at P. Hoult D.I., The principle of reciprocity in signal strength calculations - a mathematical guide , Concepts in Magn. Reson. <b>12</b>, 173-187 (2000). Stan Sykora 6th BFF 2011, Freiburg, Germany (F) Current detection of oscillating dipoles J0 = 2 κω m0/|R+i ωL| Detected signal current J0,max = 2 κm0/L is attained for R=0 ½ NJ = [4kTB w/R] Johnson noise current -½ ½ (S/N) J = J 0/N J = ωκ m0 (kTB w) R / |R+i ωL| ½ -½ -½ (S/N) J,max = m 0 ω (2kTB w) κL is attained at R= ωL Comment: While the maximum current is obtained for infinite-Q loops (R=0 ), the best S/N ratio is attained for the “matching” condition R = ωL Stan Sykora 6th BFF 2011, Freiburg, Germany (G) Voltage detection of oscillating dipoles V0 = J 0R = 2 ωκ m0 R/|R+i ωL| Detected signal voltage V0,max = 2 ωκ m0 is attained for R= ∞ ½ NV = [4kTB wR] Johnson noise voltage -½ ½ (S/N) V = V 0 /N V = ωκ m0 (kTB w) R / |R+i ωL| ½ -½ -½ (S/N) V,max = m 0 ω (2kTB w) κL is again attained at R= ωL Comment: While the maximum voltage is obtained for open loops (R = ∞), the best S/N ratio is attained for the “matching” condition R = ωL.